Frequency-wavenumber method

The contributions of 333 sources similar to the one analysed in the last section are summed together to simulate ambient vibrations. To estimate the uncertainty on the apparent velocity determination, the whole signals are split in several smaller time windows for which the array responses are computed. For each time window, the velocity of the semblance peak is searched for wavenumbers below 1.5 rad/m and for velocities between 150 and 2000 m/s. From a coarse griding in the wavenumber plane, the vector $(k_x,k_y)$ of the highest peak is iteratively refined to an arbitrary small precision. Thus, for each frequency band, an histogram of the velocities at the observed maxima is constructed (e.g. figure 6.8(a) for array C and 10 cycles). The areas below the histograms are normalized to one in the slowness domain, explaining the high values for the probability density functions. The curves in figure 6.8(c) are sections across the histograms of figures 6.8(a) and 6.8(b) at 3 Hz.

Figure 6.8: Comparison of frequency-wavenumber analysis for array C, influence of the time window length. (a) Histograms of velocities with a maximum f semblance obtained with time windows of 10 cycles. (b) Same processing with time windows of 50 cycles. (c) Cross section at 3 Hz, of the histograms of figures (a) and (b), shown by dotted and plain lines, respectively. The curves are of the same types as in figure 6.5.

The influence of the window length is first checked by calculating the histograms for time windows containing 10 and 50 cycles (figures 6.8(a) and 6.8(b), respectively) for array C. The theoretical dispersion curves are represented by the three thin plain lines. The three exponential curves (validity curves) represent constant wavenumber curves values of which correspond to the deduced $k_{min}$ (continuous line) and $k_{max}$ (dashed line). The dotted line is situated at $k_{max}/2$ . In figure 6.8(a), the average deviates from the theoretical dispersion curve with a constant bias of 50 or 100 m/s towards lower velocity, whereas all the velocity estimates are closer to the theoretical curve and the standard deviations are much smaller for the 50-cycle case (figure 6.8(b)). Both cases are calculated with the same duration of signals (six minutes), resulting in five times more windows in the 10-cycle case. To test the robustness of the statistics, one minute and 12 seconds of signals are also processed with time windows of 10 cycles, containing the same number of time windows as in the 50-cycle case calculated with the six minutes of signals. The obtained histograms are the same as in figure 6.8(a). Hence, with short time windows, increasing the number of samples neither reduces the gap to the theoretical curve nor the size of resulting error bars.

A similar processing is applied to the signals of arrays A and B (six minutes of signals and time windows of 50 cycles). The velocity histograms of arrays A, B and C can be compared in figures 6.9(a), 6.9(b) and 6.8(b), respectively. The validity curves of constant wavenumber are drawn in the same way as in figure 6.8. For all arrays, $k_{min}$ is clearly linked to the point where the velocity estimates strongly deviate from the theoretical dispersion curve shown with the thin black lines. In figure 6.8(b), bad estimations of velocity due to aliasing take place effectively between $k_{max}/2$ and $k_{max}$ . A similar conclusion could be drawn for array B, where errors towards low velocity slightly increase above $k_{max}/2$ . Due to the limited available frequency range, the aliasing effect cannot be observed for array A. Between the limits $k_{min}$ and $k_{max}/2$ , arrays B and C exhibit correct velocity estimates. For array A, the measured velocity is slightly above the theoretical Rayleigh velocity with a velocity bump between 6 and 9 Hz similar to the one of figure 6.6(a). For each array, an average and a standard deviation is calculated between $k_{min}$ and $k_{max}/2$ based on the histograms of figures 6.8(b) and 6.9. The three curves are averaged taking into account the respective weights (number of time windows) to construct the final dispersion curve plotted in figure 6.10. The measured dispersion is reliable for frequencies above 3 Hz. This limit is linked to the array sizes but also to the dramatic decrease of the noise vertical component amplitude close to the fundamental resonance frequency (2 Hz).

Figure 6.9: Results of the frequency-wavenumber method applied to arrays A (a) and B (b) with time windows including 50 cycles as in figure 6.8(b). The histograms are of the same type as at figure 6.5, the curves as well. Wavenumber limits correspond to each array geometry.

Figure 6.10: Average and standard deviations (vertical bars) of apparent dispersion curve from arrays A, B and C. The thin lines are the theoretical dispersion curves for the original ground model.

The obtained dispersion curve is inverted with five distinct runs of the neighbourhood algorithm, generating a total of 50,000 models. The parameterized model consists of a sediment layer the wave velocity of which increases with depth according to a power law, and a half-space at the base. The parameters are six: $V_p$ and $V_s/V_p$ in the two layers, the layer thickness and the $V_p$ increase between the top and the bottom of the sediment layer. Figure 6.11(a) and 6.11(b) show the velocity profiles obtained for $V_p$ and $V_s$ , respectively, for all models fitting the dispersion curve with a misfit lower than one. The misfit function is defined by equation (3.38). The dispersion curves corresponding to the misfit threshold of one are plotted in figure 6.11(c). Dispersion curve inversion leads to a good definition of the $V_s$ profile for the first 25 m. Below this depth, a large range of velocity values may explain the measured dispersion curve, due to the lack of information at low frequency. $V_p$ profile is very poorly constrained by the inversion, as $V_p$ values in the layers have very little influence on the dispersion curve for high Poisson's ratio values (section 3.1.8).

Figure 6.11: Results from inversion of the dispersion curve obtained with the frequency-wavenumber method. (a) $V_p$ , (b) $V_s$ of generated models and (c) corresponding dispersion curves. The dots and error bars represent the experimental dispersion curves to which the calculated dispersion curves are compared. The black lines of figures (a) and (b) are the velocity profiles of the true model.